extended topological quantum field theory


Functorial Quantum Field Theory


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Extended quantum field theory (or multi-tiered quantum field theory) is the fully local formulation of functorial quantum field theory, formulated in higher category theory

Whereas a

  • 1-categoricalTQFT may be regarded as a rule that allows one to compute topological invariants Z(Σ)Z(\Sigma) assigned to dd-dimensional manifolds by cutting these manifolds into a sequence {Σ i}\{\Sigma_i\} of dd-dimensional composable cobordisms with (d1)(d-1)-dimensional boundaries Σ i\partial \Sigma_i, e.g. Σ=Σ 2 Σ 1=Σ 2Σ 1\Sigma = \Sigma_2 \coprod_{\partial \Sigma_1 = \partial \Sigma_2} \Sigma_1, then assigning quantities Z(Σ i)Z(\Sigma_i) to each of these and then composing these quantities in some way, e.g. as Z(Σ)=Z(Σ 2)Z(Σ 1)Z(\Sigma) = Z(\Sigma_2)\circ Z(\Sigma_1);

we have that

  • in extended TQFT Z(Σ)Z(\Sigma) may be computed by decomposing Σ\Sigma into dd-dimensional pieces with piecewise smooth boundaries, whose boundary strata are of arbitrary codimension kk.

For that reason extended QFT is also sometimes called local or localized QFT. In fact, the notion of locality in quantum field theory is precisely this notion of locality. And, as also discussed at FQFT, this higher dimensional version of locality is naturally encoded in terms of n-functoriality of ZZ regarded as a functor on a higher category of cobordisms.


The category of extended cobordisms

The definition of a jj-cobordism is recursive. A (j+1)(j+1)-cobordism between jj-cobordisms is a compact oriented (j+1)(j+1)-dimensional smooth manifold with corners whose the boundary is the disjoint union of the target jj-cobordism and the orientation reversal of the source jj-cobordism. (The base case of the recursion is the empty set, thought of as a (1)(-1)-dimensional manifold.)

nCob dn Cob_d is an nn-category with smooth compact oriented (dn)(d-n)-manifolds as objects and cobordisms of cobordisms up to nn-cobordisms, up to diffeomorphism, as morphisms.

There are various suggestions with more or less detail for a precise definition of a higher category nCob nn Cob_n of fully extended nn-dimensional cobordisms.

A very general (and very natural) one consists in taking a further step in categorification: one takes nn-cobordisms as nn-morphisms and smooth homotopy classes of diffeomorphisms beween them as (n+1)(n+1)-morphisms. Next one iterates this; see details at (∞,n)-category of cobordisms.


Extended TQFT

Fix a base ring RR, and let CC be the symmetric monoidal nn-category of nn-RR-modules.

An nn-extended CC-valued TQFT of dimension dd is a symmetric nn-tensor functor Z:nCob dCZ: n Cob_d \rightarrow C that maps

  • smooth compact oriented dd-manifolds to elements of RR

  • smooth compact oriented (d1)(d-1)-manifolds to RR-modules

  • cobordisms of smooth compact oriented (d1)(d-1)-manifolds to RR-linear maps between RR-modules

  • smooth compact oriented (d2)(d-2)-manifolds to RR-linear additive categories

  • cobordisms of smooth compact oriented (d2)(d-2)-manifolds to functors between RR-linear categories

  • etc …

  • smooth compact oriented (dn)(d-n)-manifolds to RR-linear (n1)(n-1)-categories

  • cobordisms of smooth compact oriented (dn)(d-n)-manifolds to (n1)(n-1)-functors between RR-linear (n1)(n-1)-categories

with compatibility conditions and gluing formulas that must be satisfied… For instance, since the functor ZZ is required to be monoidal, it sends monoidal units to monoidal units. Therefore, the dd-dimensional vacuum is mapped to the unit element of RR, the (d1)(d-1)-dimensional vacuum to the RR-module RR, the (d2)(d-2)-dimensional vacuum to the category of RR-modules, etc.

Here nn can range between 00 and dd. This generalizes to an arbitrary symmetric monoidal category CC as codomain category.


Classes of examples by dimension

n=1n=1 gives ordinary TQFT.

The most common case is when R=R = \mathbb{C} (the complex numbers), giving unitary ETQFT.

The most common cases for CC are

  • C=nHilb(R)C = n Hilb(R), the category of nn-Hilbert spaces? over a topological field RR. As far as we know this is only defined up to n=2n=2.

  • C=nVect(R)C = n Vect(R), the category of nn-vector spaces over a field RR.

  • C=nMod(R)C = n Mod(R), the (conjectured?) category of nn-modules over a commutative ring RR.

3d: Turaev-Viro model

Generic examples

By the cobordism hypothesis-theorem every fully dualizable object in a symmetric monoidal (,n)(\infty,n)-category with duals provides an example.

Specific examples

See also at TCFT.


Construction of ETQFT’s

  • By generators and relations

  • By path integrals (this is Daniel Freed’s approach)

  • By modular tensor n-categories?

Classification of ETQFT’s

Assume Z:nCob dnVect(R)Z: n Cob_d \rightarrow n Vect(R) is an extended TQFT. Since ZZ maps the (d1)(d-1)-dimensional vacuum to RR as an RR-vector space, by functoriality ZZ is forced to map a dd-dimensional closed manifold to an element of RR. Iterating this argument, one is naturally led to conjecture that, under the correct categorical hypothesis, the behaviour of ZZ is enterely determined by its behaviour on (dn)(d-n)-dimensional manifolds. See details at cobordism hypothesis.

Relation of ETQFT to AQFT



More on extended QFTs is also at

Isbell duality between algebra and geometry

A\phantom{A}geometryA\phantom{A}A\phantom{A}categoryA\phantom{A}A\phantom{A}dual categoryA\phantom{A}A\phantom{A}algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand-KolmogorovAlg op\overset{\text{<a href="">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cpt\phantom{NC}TopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C *,comm op\overset{\text{<a class="existingWikiWord" href="">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}A\phantom{A}A\phantom{A}comm. C-star-algebraA\phantom{A}
A\phantom{A}noncomm. topologyA\phantom{A}A\phantom{A}NCTopSpaces H,cptNCTopSpaces_{H,cpt}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg C * op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}A\phantom{A}A\phantom{A}general C-star-algebraA\phantom{A}
A\phantom{A}algebraic geometryA\phantom{A}A\phantom{A}NCSchemes Aff\phantom{NC}Schemes_{Aff}A\phantom{A}A\phantom{A}almost by def.TopAlg fin op\overset{\text{<a href="">almost by def.</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin} A\phantom{A}A\phantom{A}fin. gen.A\phantom{A}
A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}noncomm. algebraicA\phantom{A}
A\phantom{A}NCSchemes AffNCSchemes_{Aff}A\phantom{A}A\phantom{A}Gelfand dualityTopAlg fin,red op\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}A\phantom{A}A\phantom{A}fin. gen.
A\phantom{A}associative algebraA\phantom{A}A\phantom{A}
A\phantom{A}differential geometryA\phantom{A}A\phantom{A}SmoothManifoldsSmoothManifoldsA\phantom{A}A\phantom{A}Milnor's exerciseTopAlg comm op\overset{\text{<a href="">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}A\phantom{A}A\phantom{A}commutative algebraA\phantom{A}
A\phantom{A}supergeometryA\phantom{A}A\phantom{A}SuperSpaces Cart n|q\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}A\phantom{A}A\phantom{A}Milnor's exercise Alg 2AAAA op C ( n) q\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }A\phantom{A}A\phantom{A}supercommutativeA\phantom{A}
A\phantom{A}formal higherA\phantom{A}
A\phantom{A}(super Lie theory)A\phantom{A}
ASuperL Alg fin 𝔤A\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}AALada-MarklA sdgcAlg op CE(𝔤)A\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}A\phantom{A}differential graded-commutativeA\phantom{A}
A\phantom{A} (“FDAs”)

in physics:

A\phantom{A}Poisson algebraA\phantom{A}A\phantom{A}Poisson manifoldA\phantom{A}
A\phantom{A}deformation quantizationA\phantom{A}A\phantom{A}geometric quantizationA\phantom{A}
A\phantom{A}algebra of observablesA\phantom{A}space of statesA\phantom{A}
A\phantom{A}Heisenberg pictureA\phantom{A}Schrödinger pictureA\phantom{A}
A\phantom{A}higher algebraA\phantom{A}A\phantom{A}higher geometryA\phantom{A}
A\phantom{A}Poisson n-algebraA\phantom{A}A\phantom{A}n-plectic manifoldA\phantom{A}
A\phantom{A}En-algebrasA\phantom{A}A\phantom{A}higher symplectic geometryA\phantom{A}
A\phantom{A}BD-BV quantizationA\phantom{A}A\phantom{A}higher geometric quantizationA\phantom{A}
A\phantom{A}factorization algebra of observablesA\phantom{A}A\phantom{A}extended quantum field theoryA\phantom{A}
A\phantom{A}factorization homologyA\phantom{A}A\phantom{A}cobordism representationA\phantom{A}


With an eye towards the full extension of Chern-Simons theory:

For TQFTs appearing in solid state physics in the context of topological order:

Last revised on July 3, 2016 at 02:11:09. See the history of this page for a list of all contributions to it.